Depending on which version of the game you play there will be different times and ways you catch Lugia.
If your playing Gold...
You'll find Lugia in the Whirl Islands near Cinawood City. To catch it make sure you have the Silver Wing. You get it from the old man in Pewter City after you beat the Elite Four. Your best bet with it is to use the Master Ball Prof. Elm gives you to catch it. It will be at level 70 and knows Hydro Pump, Rain Dance, Recover and Gust.
If your playing Silver...
You'll get to catch Lugia first. Make sure you have the Silver Wing before you challenge it. You can get the Silver Wing from the Radio Tower Director after you solve the Team Rocket problem. Then get HM06 Whirlpool and go to the Whirl Islands. Make sure you save your game and bring lots of Ultra Balls. Get Lugia's HP in the Red Zone and then throw Ultra Balls. It will be at level 40 and knows, Aeroblast, Recover, Gust, and Safegaurd.